The rst part of this survey is a heuristic, non-technical ... · The rst part of this survey is a heuristic, non-technical discussion of what an HHS is, and the aim is to provide

WHAT IS A HIERARCHICALLY HYPERBOLIC SPACE?

ALESSANDRO SISTO

Abstract. The first part of this survey is a heuristic, non-technical discussion of whatan HHS is, and the aim is to provide a good mental picture both to those actively doingresearch on HHSs and to those who only seek a basic understanding out of pure curiosity. Itcan be read independently of the second part, which is a detailed technical discussion of theaxioms and the main tools to deal with HHSs.

Contents

Introduction 1State of the art 2Acknowledgements 3

Part 1. Heuristic discussion 31. Standard product regions 31.1. In the examples 42. Projections to hyperbolic spaces 62.1. Distance formula and hierarchy paths 62.2. Consistency 72.3. Realisation 72.4. In the examples 8

Part 2. Technical discussion 93. Commentary on the axioms 94. Main tools 154.1. Distance formula 154.2. Hierarchy paths 154.3. Realisation 165. Additional tools 175.1. Hierarchical quasiconvexity 175.2. Hulls and their cubulation 185.3. Factored spaces 195.4. Boundary 205.5. Modifying the HHS structure 20References 20

Introduction

Hierarchically hyperbolic spaces (HHSs) were introduced in [BHS17a] as a common frame-work to study mapping class groups and cubical groups. The definition is inspired by theextremely successful Masur-Minsky machinery to study mapping class groups [MM99, MM00,Beh06]. Since [BHS17a], the list of examples has expanded significantly [BHS15, BHS17b,

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WHAT IS A HIERARCHICALLY HYPERBOLIC SPACE? 2

HS16], and the HHS framework has been used to prove several new results, including newresults for mapping class groups and cubical groups. For example, the previously knownbound for the asymptotic dimension of mapping class groups has been dramatically improvedin [BHS17b], while the main result from [BHS17c] is that top-dimensional quasi-flats in HHSsstay within bounded distance from a finite union of “standard orthants”, a fact that wasknown neither for mapping class groups nor for cubical groups without imposing additionalconstraints (see [Hua17]).

The aim of this survey article, however, is not to present the state of the art of the field,which is very much evolving. In this direction, we only give a brief description of all therelevant papers below. The main aim of this survey is, instead, to discuss the geometryof HHSs, only assuming that the reader is familiar with (Gromov-)hyperbolic spaces. Thedefinition of HHS is admittedly hard to digest if one is not presented with the geometricintuition behind it, and the aim of this survey is to remedy this shortcoming. The firstpart is aimed at the casual reader and gives a general idea of what an HHS looks like. Wewill discuss the various notions in the main motivating examples too; the reader can usewhichever example they are familiar with to gain better understanding.

The second part of the survey is mainly aimed at those who want to do research onHHSs, as well as to those who seek deeper understanding. We will discuss every axiomin detail, and then we will proceed to discuss the main tools one can use to study HHSs.Anyone who becomes familiar with the material that will be presented will have a ratherdeep understanding of HHSs. And will be ready to tackle one of the many open questionsasked in the papers we describe below...

State of the art.

‚ In [BHS17a], J. Behrstock, M. Hagen and I axiomatised the Masur-Minsky machinery,extended it to right-angled Artin groups (and many other groups acting on CAT(0)cube complexes including fundamental groups of special cube complexes), and initiatedthe study of the geometry of hierarchically hyperbolic groups by studying quasi-flatsvia “coarse differentiation”.

‚ In [BHS15] we simplified the list of axioms, which allowed us to extend the listof examples of hierarchically hyperbolic groups (and to significantly simplify theMasur-Minsky approach). The paper contains a combination theorem for trees ofHHSs, as well as other results to construct new HHSs out of old ones.

‚ Speaking of new examples, in [HS16] M.Hagen and T. Susse prove that all propercocompact CAT(0) cube complexes are HHSs.

‚ [BHS17b] deals with asymptotic dimension. We show finiteness of the asymptoticdimension of hierarchically hyperbolic groups, giving explicit estimates in certaincases. In the process we drastically improve previously known bounds on the as-ymptotic dimension of mapping class groups. We also show that many natural(small cancellation) quotients of hierarchically hyperbolic groups are hierarchicallyhyperbolic.

‚ In [DHS17], M. Durham, M. Hagen and I introduce a compactification of hierarchicallyhyperbolic groups, related to Thurston’s compactification of Teichmuller space in thecase of mapping class groups and Teichmuller space. This compactification turnsout to be very well-behaved as, for example, “quasi-convex subgroups” in a suitablesense have a well-defined and easily recognisable limit set (while the situation forTeichmuller space is more complicated). Constructing this compactification allowedus, for example, to study dynamical properties of individual elements and to prove arank rigidity result.


‚ Further study of the HHS boundary is carried out in [Mou16, Mou17], where S.Mousley shows non-existence of boundary maps in certain cases and other exoticphenomena.

‚ In [BHS17c], J. Behrstock, M. Hagen and I study the geometry of quasi-flats, that isto say images of quasi-isometric embeddings of Rn in hierarchically hyperbolic spaces.More specifically, we show that top-dimensional quasi-flats lie within finite distanceof a union of “standard orthants”. This simultaneously solves open questions andconjectures for most of the motivating examples of hierarchically hyperbolic groups,for example a conjecture of B. Farb for mapping class groups, and one by J. Brockfor the Weil-Petersson metric, and it is new in the context of CAT(0) cube complexestoo.

‚ In [Spr], D. Spriano studies non-trivial HHS structures on hyperbolic spaces, and usesthem to show that certain natural amalgamated products of hierarchically hyperbolicgroups are hierarchically hyperbolic.

‚ In [ABD17], C. Abbott, J. Behrstock and M. Durham prove that hierarchicallyhyperbolic groups admit a “best” acylindrical action on a hyperbolic space, andprovide a complete classification of stable subgroups of hierarchically hyperbolicgroups.

‚ In [Hae16], T. Haettel studies homomorphisms of higher rank lattices to hierarchicallyhyperbolic groups, finding severe restrictions.

‚ In [ST16], using part of an HHS structure, S. Taylor and I studied various notions ofprojections, including subsurface projection for mapping class groups, along a randomwalk, and used this to prove a conjecture of I. Rivin on random mapping tori.

Acknowledgements. This article has been written for the proceedings for the “Beyondhyperbolicity” conference held in June 2016 in Cambridge, UK. The author would like tothank Mark Hagen, Richard Webb, and Henry Wilton for organising the conference and thewonderful time he had in Cambridge.

The author would also like to thank Jason Behrstock, Mark Hagen, and Davide Sprianofor useful comments on previous drafts of this survey.

Part 1. Heuristic discussion

1. Standard product regions

In this section we discuss the first heuristic picture of an HHS, which is the one providedby standard product regions.

If an HHS X is not hyperbolic, then the obstruction to its hyperbolicity is encoded by thecollection of its standard product regions. These are quasi-isometrically embedded subspacesthat split as direct products, and the crucial fact is that each standard product region, aswell as each of its factors, is an HHS itself, and in fact an HHS of lower “complexity”. Itis not very important at this point, but the complexity is roughly speaking the length of alongest chain of standard product regions P1 Ĺ P2 Ĺ ¨ ¨ ¨ Ĺ Pn contained in the HHS; what isimportant right now is that factors of standard product regions are “simpler” HHSs, and the“simplest” HHSs are hyperbolic spaces. This is what allows for induction arguments, wherethe base case is that of hyperbolic spaces.

Standard product regions encode entirely the non-hyperbolicity of the HHS X in thefollowing sense. Given a, say, length metric space pZ, dq and a collection of subspaces P , onecan define the cone-off of Z with respect to the collection of subspaces (in several differentways that coincide up to quasi-isometry, for example) by setting d1px, yq :“ 1 for all x, ycontained in the same P P P and d1px, yq “ dpx, yq otherwise, and declaring the cone-off


distance between two points x, y to be infx“x0,...,xn“yř

d1pxi, xi`1q. This has the effect ofcollapsing all P P P to bounded sets, and the reason why this is a sensible thing to do is thatone might want to consider the geometry of Z “up to” the geometry of the P P P. When Zis a graph, as is most often the case for us, coning-off amounts to adding edges connectingpairs of vertices contained in the same P P P.

Back to HHSs, when coning-off all standard product regions of an HHS one obtains ahyperbolic space, that we denote CS1. In other words, an HHS is weakly hyperbolic relativeto the standard product regions. Roughly speaking, when moving around X , one is eithermoving in the hyperbolic space CS or in one of the standard product regions. The philosophybehind many induction arguments for HHSs is that when studying a certain “phenomenon”,either it leaves a visible trace in CS, or it is “confined” in a standard product region, andcan hence be studied there. For example, if the HHS is in fact a group, one can consider thesubgroup generated by an element g, and it turns out that either the orbit maps of g in CSare quasi-isometric embeddings, or g virtually fixes a standard product region [DHS17].

So far we discussed the “top-down” point of view on standard product regions, but there isalso a “bottom-up” approach. In fact, one can regard HHSs as built up inductively startingfrom hyperbolic spaces, in the following way:

‚ hyperbolic spaces are HHSs,‚ direct products of HHSs are HHSs,‚ “hyperbolic-like” arrangements of HHSs are HHSs.

The third bullet refers to CS being hyperbolic, and the fact that CS can also be thoughtof as encoding the intersection pattern of standard product regions. Incidentally, I believethat there should be a characterisation of HHSs that looks like the list above, i.e. that bysuitably formalising the third bullet one can obtain a characterisation of HHSs. This has notbeen done yet, though. There is, however, a combination theorem for trees of HHSs in thisspirit [BHS15].

One final thing to mention is that standard product regions have well-behaved coarseintersections, meaning that the coarse intersection of two standard product regions is well-defined and coarsely coincides with some standard product region. In other words, X isobtained gluing together standard product regions along sub-HHSs, so a better version ofthe third bullet above would be “hyperbolic-like arrangements of HHS glued along sub-HHSsare HHSs”.

1.1. In the examples. We now discuss standard product regions in motivating examples ofHHSs.

RAAGs. Consider a simplicial graph Γ. Whenever one has a (full) subgraph Λ of Γ which isthe join of two (full, non-empty) subgraphs Γ1,Γ2, then the RAAG AΓ contains an undistortedcopy of the RAAG AΛ « AΛ1ˆAΛ2 . Such subgroups and their cosets are the standard productregions of AΓ. In this case, CS is a Cayley graph of AΓ with respect to an infinite generatingset (unless Γ consists of a single vertex), namely V Γ Y tAΛ ă AΓ : Λ “ joinpΛ1,Λ2qu. Agiven HHS can be given different HHS structures (which turns out to allow for more flexibilitywhen performing various constructions, rather than being a drawback), and one instanceof this is that one can regard as standard product regions all AΛ where Λ is any propersubgraph of Γ, one of the factors being trivial. In this case CS is the Cayley graph of AΓ withrespect to the generating set V ΓY tAΛ ă AΓ : Λ proper subgraph of Γu, which is perhapsmore natural.

1This notation is taken from the mapping class group context, even though it’s admittedly not the bestnotation in other examples.


For both HHS structures described above, CS is not only hyperbolic, but in fact quasi-isometric to a tree.

Mapping class groups. Given a surface S, there are some “obvious” subgroups of MCGpSqthat are direct products. In fact, consider two disjoint (essential) subsurfaces Y, Z of S. Anytwo self-homeomorphisms of S supported respectively on Y and Z commute. This yields(up to ignoring issues related to the difference between boundary components and puncturesthat I do not want to get into) a subgroup of MCGpSq isomorphic to MCGpY q ˆMCGpZq.Such subgroups are in fact undistorted. One can similarly consider finitely many disjointsubsurfaces instead, and this yields the standard product regions in MCGpSq. More precisely,one should fix representatives of the (finitely many) topological types of collections of disjointsubsurfaces, and consider the cosets of the subgroups as above. In terms of the markinggraph, product regions are given by all markings containing a given sub-marking.

In this case it shouldn’t be too hard to convince oneself that CS as defined above isquasi-isometric to the curve complex, see [MM99, ]. To re-iterate the philosophy explainedabove, if some behaviour within MCGpSq is not confined to a proper subsurface Y , thenthe geometry of CS probably comes into play when studying it, and otherwise it is mostconvenient to study the problem on the simpler subsurface Y .

CAT(0) cube complexes. Hyperplanes are crucial for studying CAT(0) cube complexes, andthe carrier of a hyperplane (meaning the union of all cubes that the hyperplane goes through)is naturally a product of the hyperplane and an interval. It is then natural, when trying todefine an HHS structure on a CAT(0) cube complex, to regard carriers of hyperplanes asstandard product regions, even though one of the factors is bounded.

This is not enough, though. As mentioned above, coarse intersections of standard productregions should be standard product regions. But it is easy to describe the coarse intersectionof two carriers of hyperplanes, or more generally the coarse intersection of two convexsubcomplexes. Given a convex subcomplex Y of the CAT(0) cube complex X , one canconsider the gate map gY : X Ñ Y , which is the closest-point projection in either the CAT(0)

or the `1–metric (they coincide). More combinatorially, for x P X p0q, gY pxq is defined bythe property that the hyperplanes separating x from gY pxq are exactly those separating xfrom Y . The coarse intersection of convex subcomplexes Y, Z is just gY pZq, which is itself aconvex subcomplex.

Back to constructing an HHS structure on a CAT(0) cube complex, we now know that weneed to include as standard product regions all gates of carriers in other carriers. But thenwe are not done yet, because for the same reason that we need to take gates of carriers wealso need to take gates of gates, and so on. Also, we need this process to stabilise eventually(which is not always the case, unfortunately), because an HHS needs to have finite complexityto allow for induction arguments. All these considerations lead to the definition of factorsystem. Rather than carrier of hyperplanes, we will consider combinatorial hyperplanes,which are the two copies of a hyperplane that bound its carrier, but this does not make asubstantial difference for the purposes of this discussion.

Definition 1.1. A factor system F for the cube complex X is a collection of convexsubcomplexes so that:

(1) all combinatorial hyperplanes are in F ,(2) there exists ξ ě 0 so that if F, F 1 P F and gF pF

1q has diameter at least ξ, thengF pF

1q P F .(3) F is uniformly locally finite.

Any factor system F on a cube complex gives an HHS structure, where CS is obtainedconing off all members of F . It turns out that CS is quasi-isometric to a tree. It is proven in


[HS16] that all cube complexes admitting a proper cocompact action by isometries have afactor system, and are therefore HHSs.

2. Projections to hyperbolic spaces

In this section we discuss a point of view on HHSs that is more similar to the actualdefinition, and it is in terms of “coordinates” in certain hyperbolic spaces.

We already saw that any HHS X comes equipped with a hyperbolic space, CS, obtainedcollapsing the standard product regions. In particular, there is a coarsely Lipschitz mapπS : X Ñ CS.

The (coarse geometry of the) hyperbolic space CS is not enough to recover the wholegeometry of X , since it does not contain information about the standard product regionsthemselves. Hence, we want something to keep track of the geometry of the standard productregions. Since factors of standard product regions are HHSs themselves, they also comewith a hyperbolic space obtained collapsing the standard product sub-regions. Consideringall standard product regions, we obtain a collection of hyperbolic space tCY uY PS, which,together, control the geometry of X , as we are about to discuss. The index set S is the set offactors of standard product regions, where the whole of X should be considered as a productregion with (a trivial factor and the other factor being) S P S, so as to include CS amongthe hyperbolic spaces we consider.

Another piece of data we need is a collection of coarsely Lipschitz maps πY : X Ñ CY forall Y P S, which allow us to talk about the geometry of X “from the point of view of CY ”.These projection maps come from natural coarse retractions of X onto the standard productregions, composed with the collapsing maps, but for now it only matters that the πY exist.

2.1. Distance formula and hierarchy paths. The first way in which the CY control thegeometry of X is that whenever x, y P X are far away, then their projections to some CY arefar away, so that any coarse geometry feature of X leaves a trace in at least one of the CY .In fact, there is much better control on distances in X in terms of distances in the variousCY , and this is given by the distance formula. This is perhaps the most important piece ofmachinery in the HHS world, and certainly the most iconic. To state it, we need a littlebit of notation. We write A «K B if A{K ´K ď B ď KB `K, and declare ttAuuL “ A ifA ě L, and ttAuuL “ 0 otherwise. The distance formula says that for all sufficiently large Lthere exists K so that

dX px, yq «Kÿ

Y PS

ttdCY pπY pxq, πY pyqquuL

for all x, y P X . In words, the distance in X between two points is, up to multiplicativeand additive constants, the sum of the distances between their far-away projections in thevarious CY . Very imprecisely, this is saying that X quasi-isometrically embeds in

ś

Y PS CYendowed with some sort of `1 metric. To save notation one usually writes dY px, yq instead ofdCY pπY pxq, πY pyqq.

Another important fact related to the distance formula is the existence of hierarchy paths,that is to say quasigeodesics in X that shadow geodesics in each CY . Namely, there existsD so that for any x, y P X there exists a pD,Dq–quasigeodesic γ joining them so thatπY ˝ γ is an unparametrised pD,Dq–quasigeodesic in CY . Since CY is hyperbolic, beingan unparametrised quasigeodesic means that (the image of) πY ˝ γ is Hausdorff-close to ageodesic, and it “traverses” the geodesic coarsely monotonically. In most cases it is muchbetter to deal with hierarchy paths than with geodesics.


2.2. Consistency. The distance formula alone is not enough for almost anything, but thepoint is that it comes with a toolbox that one uses to control the various projection termsby constraining certain projections in terms of certain other projections. In Part 2, we willanalyse these tools in detail. For now we will instead describe what happens to variousprojections when moving along a hierarchy path, which gives the right picture about howprojections are constrained.

The tl; dr version of this subsection is: for certain pairs Y, Z P S, along a hierarchypath one can only change the projections to Y,Z in a specified order. This is sufficient tounderstand most of the next subsection.

Nesting. Let x, y P X and suppose that dY px, yq is large. Notice that Y (which is a factorof a standard product region) gives a bounded set in CS (which was obtained from X bycollapsing standard product regions), which we denote ρYS . We know that when moving alongany hierarchy path γ from x to y, the projection to Y needs to change. This is how thishappens: γ has an initial subpath where the projection to Y coarsely does not change, whilethe projection to CS approaches ρYS . All the progress that needs to be made by γ in CY ismade by a middle subpath whose projection to CS remains close to ρYS . Then, there is a finalsubpath that does not make any progress in CY and takes us from ρYS to πSpyq in CS. Inshort, you can only make progress in CY if you are close to ρYS in CS.

The description above applies to more general pairs of elements of S, namely wheneverS is replaced by some Z so that Y is properly nested into Z, denoted Y Ĺ Z. Nesting justmeans that the factor Y is contained in the factor Z, or more precisely that there is a copyof Y in a standard product region that is contained in a copy of Z.

Orthogonality. We just saw that changing projections in both Y and Z when Y Ĺ Z canonly be done in a rather specific way. The opposite situation is when Y and Z (which, recall,are factors of standard product regions) are orthogonal, denoted Y KZ, meaning that theyare (contained in) different factors of the same product region. In this case, along a hierarchypath there is no constraint regarding which projection needs to change first, and in fact theycan also change simultaneously. Ortohogonality is what creates non-hyperbolic behaviour inHHSs, and is what one has to constantly fight against.

Transversality. When Y,Z are neither Ď- nor K-comparable, we say that they are transverse.This is the generic case. When Y&Z and x, y P X are so that dY px, yq, dZpx, yq are bothlarge, up to switching Y,Z what happens is the following. When moving along any hierarchypath from x to y one has to first change the projection to CY until it coarsely coincides withπY pzq, and only then the projection to CZ can start moving from πZpxq to πZpyq.

Arguably the most useful feature of transversality is a slight generalisation of this. Givenx, y P X , and a set Y Ď S of pairwise transverse elements so that dY px, yq is large for everyY P Y, there is a total order on Y so that, whenever Y ă Z, along any hierarchy path fromx to y the projection to Y has to change before the projection to Z does, as described above.

2.3. Realisation. Even though we did not formally describe them, we saw that for certainpairs Y, Z P S, namely when Y & Z, there are some constraints on the projections of pointsin X to CY, CZ. These are called consistency inequalities. As it turns out, the consistencyinequalities are the only obstructions for “coordinates” pbY P CY qY PS to be coarsely realisedby a point x in X , meaning that πY pxq coarsely coincide with bY in each CY . This isimportant because it allows to perform constructions in each of the CY separately and thenput everything back together.

To make this principle clear, we now give an example of a construction of this type. Say wewant to construct a “coarse median” map m : X 3 Ñ X (in the sense of [Bow13]), which let’sjust take to mean a coarsely Lipschitz map so that mpx, x, yq is coarsely x. Consider x, y, z


in X , and let us define mpx, y, zq by defining its coordinates in the CY . Given Y P S, thetriangle with vertices πY pxq, πY pyq, πY pzq has a coarse centre bY , because CY is hyperbolic.It turns out that the coordinates pbY q satisfy the consistency inequalities, so that one candefine mpx, y, zq as the realisation point. As an aside, it is a nice exercise to use the propertiesof m to show that X satisfies a quadratic isoperimetric inequality.

To sum up, the distance formula says that the natural map X Ñś

Y PS CY is “coarselyinjective”, and the consistency inequalities provide a coarse characterisation of the image.

2.4. In the examples.

RAAGs. In the case of RAAGs, S (the set of factors of product regions) is the set of cosetsof sub-RAAGs, considered up to parallelism. We say that gAΛ, hAΛ Ď AΓ are parallel ifg´1h commutes with every element of AΛ, which essentially means that there’s a productgpAΛˆ ă g´1h ąq inside the RAAG AΛ so that gAΛ, hAΛ are copies of one of the factors.Taking parallelism classes ensures that we will not do multiple counting in the distance formula.What we mean is that infinitely many parallel cosets would give the same contribution tothe distance formula, which would clearly break it.

As in the case of CS, CpgAΛq is a copy of the Cayley graph of AΛ with respect to thegenerating set V ΛY tAΛ1 ă AΛ : Λ1 proper subgraph of Λu. The projection map from AΓ toCpgAΓq is the composition of the closest-point projection to gAΛ in the usual Cayley graphof AΛ, and the inclusion gAΛ Ď CpgAΓq. The closest-point projection can also be rephrasedin terms of the normal form for elements of AΓ, since the normal form gives geodesics.

Nesting is inclusion up to parallelism, meaning that we declare rgAΛs Ď rgAΛ1s whenΛ Ď Λ1, where r¨s denotes the parallelism class. Similarly, we declare rgAΛsKrgAΛ1s if Λ,Λ1

form a join.In the case of RAAGs, it turns out that geodesics in (the usual Cayley graph of AΛ) are

actually hierarchy paths.

Mapping class groups. In this case, S is the collection of (isotopy classes of essential)subsurfaces, with each CY being the corresponding curve complex, and the maps πY aredefined using the so-called subsurface projections. Nesting is containment (up to isotopy),while orthogonality corresponds to disjointness (again up to isotopy).

CAT(0) cube complexes. Consider a CAT(0) cube complex X with a factor system F . In thiscase, S is the union of tS “ X u and the set of parallelism classes in F . Parallelism can bedefined in at least two equivalent ways. The first one is that the convex subcomplexes F, F 1

are parallel if they cross the same hyperplanes. The second one, which provides a much betterpicture, is that F, F 1 are parallel if there exists an isometric embedding of F ˆ r0, ns Ñ X ,where r0, ns is cubulated by unit intervals and F ˆ r0, ns is regarded as a cube complex, sothat F ˆ t0u maps to F in the obvious way, and the image of F ˆ tnu is F 1. As in the caseof RAAGs, if we did not take parallelism classes then the distance formula would certainlynot work due to multiple counting.

The CrF s are obtained starting from F and coning off all F 1 P F contained in F . Themaps πrF s are defined using gates.

Nesting rF s Ď rF 1s is inclusion up to parallelism, which can also be rephrased as: allhyperplanes crossing F also cross F 1 (notice that this does not depend on the choice ofrepresentatives). Orthogonality rF sKrF 1s means that, up to parallelism, F ˆF 1 has a naturalembedding into X . It can also be rephrased as: each hyperplane crossing F crosses eachhyperplane crossing F 1.


Part 2. Technical discussion

Keeping in mind the heuristic discussion from Part 1, we now analyse in more detail thedefinition and the main tools to study HHSs. We start with the axioms.

We will often motivate the axioms in terms of standard product regions, but we warn thereader in advance that those will be constructed only after we discuss all the axioms and afew tools. This, however, is inevitable. In fact, we are trying to describe a space that hassome sort of subspaces, the standard product regions, that can be endowed with the samestructure as the space itself. Until we know what that structure is in detail, we cannot use itto construct the standard product regions starting from first principles. Hopefully, one ormore of the examples we discussed in Part 1 can help with intuition.

3. Commentary on the axioms

We will work in the context of a quasigeodesic space, X , i.e., a metric space where anytwo points can be connected by a uniform-quality quasigeodesic. It is more convenient for usto work with quasi-geodesic metric spaces than geodesic metric space because the standardproduct regions are in a natural way quasi-geodesic metric spaces, rather than geodesicmetric space. Any quasi-geodesic metric space is quasi-isometric to a geodesic metric spacesince one can consider an approximating graph whose vertices form a maximal net, so forthe purposes of large-scale geometry there’s basically no difference between geodesic andquasi-geodesic metric spaces.

Actually, all the requirements in the definition of HHS are meant to be stable under passingto standard product regions. We do not have standard product regions yet, so what happensin the definition instead is that the axioms are about certain sub-collections of the set ofhyperbolic spaces involved in the HHS structure, rather than just the whole collection.

We now go through the definition of HHS given in [BHS15], which is the one with “optimised”axioms compare to [BHS17a]. The statements of the axioms are given exactly as in [BHS15].

The q–quasigeodesic space pX , dX q is a hierarchically hyperbolic space if there exists δ ě 0,an index set S, and a set tCW : W P Su of δ–hyperbolic spaces pCU, dU q, such that thefollowing conditions are satisfied:

(1) (Projections.) There is a set tπW : X Ñ 2CW | W P Su of projections sendingpoints in X to sets of diameter bounded by some ξ ě 0 in the various CW P S.Moreover, there exists K so that each πW is pK,Kq–coarsely Lipschitz.

The index set S is the set of factors of standard product regions. Any V P S hencecorresponds to each of many “parallel” subsets of X . We already saw where the hyperbolicspaces associated to an HHS comes from: each factor of a standard product region containsvarious standard product sub-regions, which we can cone-off to obtain a hyperbolic space.The way to think about the projection is that the standard product regions and their factorscome with a coarse retraction from X , and the projections πW in the definition are thecomposition of those retractions with the cone-off map. This is admittedly a bit circularbecause we will later define the retractions in terms of the πW , but should hopefully help tounderstand the picture.

The reason why the projections take value in bounded subsets of the CW rather thanpoints is just that in several situations, for example subsurface projections for mapping classgroups, this is what one gets in a natural way. One can make arbitrary choices and modifythe projections to take value in points, and nothing would be affected.


(2) (Nesting.) S is equipped with a partial order Ď, and either S “ H or S contains aunique Ď–maximal element; when V Ď W , we say V is nested in W . We require thatW Ď W for all W P S. For each W P S, we denote by SW the set of V P S such thatV Ď W . Moreover, for all V,W P S with V Ĺ W there is a specified subset ρVW Ă CWwith diamCW pρ

VW q ď ξ. There is also a projection ρWV : CW Ñ 2CV . (The similarity in

notation is justified by viewing ρVW as a coarsely constant map CV Ñ 2CW .)

Nesting corresponds to inclusion between standard product regions. The maximal elementcorresponds to X itself, thought of as a product region with a trivial factor.

Recall that the CW are obtained coning-off standard product regions, i.e. making thembounded. For V Ĺ W , the bounded set ρVW is one such bounded set, where V is regardedas a standard product region with one trivial factor. In the other direction, ρWV is obtainedrestricting the retraction to W .

We will discuss below the fact that ρWV for V Ĺ W is not strictly needed, and can be to alleffects and purposes be reconstructed from πW and πV .

Regarding the notation, the ρVW s in this axiom as well as the ones below always go “fromtop to bottom”, meaning that ρVW is always some kind of projection from V to W .

(3) (Orthogonality.) S has a symmetric and anti-reflexive relation called orthogonality:we write V KW when V,W are orthogonal. Also, whenever V Ď W and WKU , werequire that V KU . Finally, we require that for each T P S and each U P ST for whichtV P ST : V KUu ‰ H, there exists W P ST ´ tT u, so that whenever V KU and V Ď T ,we have V Ď W . Finally, if V KW , then V,W are not Ď–comparable.

Orthogonality is what creates non-trivial products: V and W are orthogonal if theyparticipate in a common standard product region, meaning that they are distinct factors.With this interpretation, it should be clear why when V Ď W and WKU , we require V KU ,and also why when V KW , then V,W should not be Ď–comparable.

The tricky part is the one about tV P ST : V KUu. Let us first discuss the case of S insteadof more general ST . The point is that one wants to define an orthogonal complement of theV P S, and one wants it to be an HHS, with corresponding index set UK “ tU P S : V KUu.For that to be the case, one would want UK to contain a Ď–maximal element (if it isnon-empty). The axiom says something a bit weaker, because the W containing each V KUis not required to be itself orthogonal to U . This is still enough to have an HHS structureon the orthogonal complement. The only reason we did not require the stronger versionwith W K U in [BHS17a] is that at the time we were not able to prove that such W existsin the case of CAT(0) cube complexes. However, it follows from [HS16, Theorem 3.5] thatproper cocompact cube complexes satisfy the stronger version of the axiom, so that in fact allnatural examples of HHS (so far) do, and there is no harm in strengthening the orthogonalityaxiom. In fact, sometimes the weaker formulation gives technical problems.

As a final comment, it is natural to formulate the axiom for general ST instead of just forS because all axioms need to work inductively for product regions.

(4) (Transversality and consistency.) If V,W P S are not orthogonal and neither isnested in the other, then we say V,W are transverse, denoted V&W . There exists κ0 ě 0such that if V&W , then there are sets ρVW Ď CW and ρWV Ď CV each of diameter at mostξ and satisfying:

min

dW pπW pxq, ρVW q, dV pπV pxq, ρ

WV q

(

ď κ0


for all x P X .For V,W P S satisfying V Ď W and for all x P X , we have:

min

dW pπW pxq, ρVW q, diamCV pπV pxq Y ρ

WV pπW pxqqq

(

ď κ0.

The preceding two inequalities are the consistency inequalities for points in X .Finally, if U Ď V , then dW pρ

UW , ρ

VW q ď κ0 whenever W P S satisfies either V Ĺ W or

V&W and W & U .

Transversality is best thought of as being in “general position”. As an aside for the readerwho speaks relative hyperbolicity, if U&V , then they behave very similarly to distinct cosetsof peripheral subgroups of a relatively hyperbolic group; for example the projections toU, V should be compared to closest-point projections onto a pair of distinct cosets. Thefirst consistency inequality, also known as Behrstock inequality, is very important, so wenow discuss a few ways to think about it (and its consequences). Incidentally, we note thatthe Behrstock inequality is important beyond the HHS world too; for example, it plays aprominent role in the context of the projection complexes from [BBF15], which have manyapplications.

In words, the Behrstock inequality says that if V&W and x P X projects far from ρWV inCV , then x projects close to ρVW in CW . (ρWV is best thought of as the projection of W ontoCV .) Let us start by discussing an easy situation where the inequality holds. Suppose that Vand W are two quasi-convex subsets of a hyperbolic space and suppose that πV pW q, πW pV qare both bounded, where πV , πW denote (coarse) closest point projections. Then settingρVW “ πV pW q and ρWV “ πW pV q, the first consistency inequality holds, and it is illustrated inthe following picture:

Figure 1. The Behrstock inequality for quasiconvex subspaces of a hyperbolic space.

Here is a sketch of the argument, which should also clarify the meaning of the inequality.If x projects to V far away from the projection of W , as in the picture, the we have to showthat it projects close to the projection to V onto W . This is because any geodesic from x toW must pass close to V , by a standard hyperbolicity argument.

This last fact is useful to keep in mind: in the situation above, to go from x to W one hasto pass close to V first, and change the projection to V in the process.

A second way to understand the inequality is to draw the image of π “ πV ˆ πW , which iscoarsely the following “cross”:


Figure 2. The image of π “ πV ˆ πW , when V&W .

From this graph we see a similar phenomenon to the one above: depending on whereπpxq, πpyq lie on the cross, to go from x to y one has to change the projection to V first orthe projection to W first.

This brings us to an important consequence of the Behrstock inequality, which is that onecan order transverse V,W that lie “between” x and y. Suppose that x, y P X and tViu arepairwise transverse and so that dVipx, yq are all much larger than the constant in the Behrstockinequality. Then for each V,W P tViu, up to switching V,W , the situation looks like Figure 2,and in this case we write V ă W . We can give several equivalent description of “the pictureabove”, and manipulating the Behrstock inequality reveals that they are all equivalent. Theseare the following, where we are assuming dV pπV pxq, πV pyqq, dW pπW pxq, πW pyqq ě 10E forsome sufficiently large E:

‚ V ă W ,‚ dW pπW pxq, ρ

VW q ď E,

‚ dW pπW pyq, ρVW q ą E,

‚ dV pπV pxq, ρWV q ą E,

‚ dV pπV pyq, ρWV q ď E.

A very important fact is that ă is a total order on tViu. My favourite way to draw this isthe following, assuming for simplicity Vi ă Vj if and only if i ă j:

This picture does not really take place anywhere, but it contains interesting information.You can pretend that the CVi are quasiconvex subsets of a hyperbolic space as in Figure 1,with the path from x to y in the picture representing a geodesic from x to y that passes closeto them in the order given by ă. From the picture you can read off where the various ρs are

by following the path. In particular, you see that for i ă j ă k, ρVjVi

and ρVkVi coarsely coincide

with each other and with πV pyq. This picture still works to understand where projections lieif you, for example, add another point z. You can try to convince yourself, first from thepicture and then formally, that if z projects “in the middle” on some CVi then, for j ą i,πVj pzq coarsely coincides with πVj pxq.


We now discuss the second consistency inequality in conjunction with another axiom:

(7) (Bounded geodesic image.) For all W P S, all V P SW ´ tW u, and all geodesics γof CW , either diamCV pρ

WV pγqq ď E or γ XNEpρ

VW q ‰ H.

In words, when V is properly nested into W , then the projection ρWV from CW to CV iscoarsely constant along geodesics far from ρVW (recall that this is the copy of V that getsconed-off to make CW out of W ).

This is virtually always used together with the second consistency inequality, which impliesthat if πW pxq is far from ρVW for some x P X then ρWV pπW pxqq coarsely coincides with πV pxq.This yields the version of bounded geodesic image that most often gets used in practice:

Lemma 3.1 (See e.g. [BHS17c, Lemma 1.5]). Let pX,Sq be hierarchically hyperbolic. Up toincreasing E as in the bounded geodesic image axiom, for all W P S, all V P SW ´ tW u,and all x, y P X so that some geodesic from πW pxq to πW pyq stays E–far from ρVW , we havedV pπV pxq, πV pyqq ď E.

Figure 3. In the picture we have V Ĺ W and γ is a geodesic. According tobounded geodesic image, πV pxq and πV pyq coarsely coincide.

One can simply replace the bounded geodesic image axiom and the second consistencyinequality with the lemma, since ρWV can be reconstructed from πW and πV at least onπW pX q in view of the lemma. However, for some purposes one still needs ρWV . This is mostnotably the case for the realisation theorem.

Another picture to keep in mind regarding bounded geodesic image is that, given x, y P Xand W , one can consider all V Ĺ W so that dV pπV pxq, πV pyqq is large. The correspondingρVW will form a “halo” around a geodesic from πW pxq to πW pyq.

(5) (Finite complexity.) There exists n ě 0, the complexity of X (with respect to S), sothat any set of pairwise–Ď–comparable elements has cardinality at most n.


This axiom should be pretty self-explanatory. Induction on complexity is very common inthe HHS world. The base case (complexity 1) is that of hyperbolic spaces.

(6) (Large links.) There exists E ě maxtξ, κ0u such that the following holds. Let W P Sand let x, x1 P X . Let N “ dW pπW pxq, πW px

1qq. Then, there exist T1, . . . , TtNu P SW ´

tW u such that for all T P SW ´tW u, either T P STi for some i, or dT pπT pxq, πT px1qq ă

E. Also, dW pπW pxq, ρTiW q ď N for each i.

In words, the axioms say that, given W and x, x1 P X , each of the V Ĺ W so thatdV pπV pxq, πV px

1qq is nested into one of a few fixed Ti Ĺ W . The number of Ti required isbounded only in terms of dW pπW pxq, πW px

1qq (which can be much smaller than their distancein X ).

This axiom is very related to bounded geodesic image, and in fact in concrete examplesthey are often proven at the same time. Bounded geodesic image provides a “halo” of ρVWaround a geodesic connecting πW pxq, πW px

1q, and there can be arbitrarily many of these.However, large links organises them into a few (possibly intersecting) subsets, each of whichcontains the ρVW with V nested into some fixed Ti. The number of such Ti is bounded interms of the distance dW pπW pxq, πW px

1qq.Large links is used in arguments of the following type. Consider two points x, y that are

far in X . If they are far in CS (meaning that their projections are), then one can use thegeometry of CS to study whatever property one is interested in. Otherwise, there are fewTis, and one can then analyse corresponding standard product regions. In one of them, (theretractions of) x, y are still far away, so one can use induction based on the fact that thestandard product region is an HHS of strictly lower complexity.

One concrete lemma that makes this more precise is the ”passing up” lemma [BHS15,Lemma 2.5]. This says the (contrapositive of the) following. If one has x, y P X and someSi P S so that the dSipπSipxq, πSipyqq are all large and each Si is Ď–maximal with thisproperty, then there is a bound on how many Si there are.

(8) (Partial Realization.) There exists a constant α with the following property. Let tVjube a family of pairwise orthogonal elements of S, and let pj P πVj pX q Ď CVj. Then thereexists x P X so that:‚ dVj px, pjq ď α for all j,

‚ for each j and each V P S with Vj Ĺ V , we have dV px, ρVjV q ď α, and

‚ if W&Vj for some j, then dW px, ρVjW q ď α.

Roughly speaking, the axiom says that, given pairwise-orthogonal tViu there is no restrictionon the projections of points of X to the CVi; any choice of coordinates can be realised bya point in X . This is the opposite of what happens when V & W (i.e. in one of the casesV “ W,V Ĺ W,W Ĺ V or V&W ), in which case there are serious restrictions on theprojections in view of the consistency inequalities.

This axiom gives us the first glimpse of how the standard product regions arise, and whattheir coordinates in the various CU look like. Starting from the family of pairwise orthogonalelements tVju, we see that the axioms provides us with the freedom to move independently ineach of the CVj . When we will have the “full” realisation theorem, this will give us a productregion associated to tVju. The second condition can be explained as follows: the coordinatein CV does not coarsely vary when moving around the standard product region because thestandard product region is coned-off there. The third condition tells us that “generic” pairs


of standard product regions do not interact much with each other. (Recall that we think oftransversality as being in “generic position”.)

(9) (Uniqueness.) For each κ ě 0, there exists θu “ θupκq such that if x, y P X anddpx, yq ě θu, then there exists V P S such that dV px, yq ě κ.

Informally, the axiom says that if x, y are close in each CV (meaning that their projectionsare) then x, y are close in X .

This axiom is a weaker form of distance formula. The point is that it is in many circum-stances much easier to prove than the “full” distance formula, allowing for easier proofs thatcertain spaces are HHS. This is the case for mapping class groups, where there’s a one-pageargument for this axiom, given in [BHS15, Section 11], while the known proofs of the distanceformula are much more involved.

4. Main tools

In addition to the axioms, there are 3 fundamental properties of HHSs. These were actuallypart of the first set of axioms, but they have a much higher level of sophistication than anyof the axioms.

4.1. Distance formula. We stated the distance formula in Part 1, but let us recall it. GivenA,B P R, the symbol ttAuuB will denote A if A ě B and 0 otherwise. Given C,D, we writeA —C,D B to mean C´1A´D ď B ď CA`D.

To save notation, we denote dW px, yq “ dW pπW pxq, πW pyqq.

Theorem 4.1 (Distance Formula, [BHS15, Theorem 4.5]). Let pX,Sq be hierarchicallyhyperbolic. Then there exists s0 such that for all s ě s0 there exist constants K,C such thatfor all x, y P X ,

dX px, yq —K,Cÿ

WPS

ttdW px, yquus .

The distance formula allows one to reconstruct the geometry of X from that of thehyperbolic spaces CW , and at this point its importance should hopefully be evident. It isimportant to note that the distance formula works for any sufficiently high threshold. This isuseful in practice because typically one proceeds along the following lines. One starts with aconfiguration in X , projects it to the CW and keeps into account the distance formula tofigure out what one gets. Then one performs some coarse construction in the CW , and thengoes back to X . In the process, more often than not some projections gets moved a boundedamount. To compensate for this, one uses a higher threshold in the distance formula.

4.2. Hierarchy paths. Hierarchy paths are quasigeodesics in X that shadow geodesics inall CW , which is clearly a very nice property to have since we want to relate the geometry ofX to that of the CW . Let us define them precisely.

For M a metric space, a (coarse) map f : r0, `s ÑM is a pD,Dq–unparameterized quasi-geodesic if there exists a strictly increasing function g : r0, Ls Ñ r0, `s such that f ˝g : r0, Ls ÑM is a pD,Dq–quasigeodesic and for each j P r0, LsXN, we have diamM pfpgpjqq Y fpgpj ` 1qqq ďD.

Definition 4.2 (Hierarchy path). Let pX,Sq be hierarchically hyperbolic. For D ě 1, a(not necessarily continuous) path γ : r0, `s Ñ X is a D–hierarchy path if

(1) γ is a pD,Dq-quasigeodesic,(2) for each W P S, the path πW ˝ γ is an unparameterized pD,Dq–quasigeodesic.


Theorem 4.3 (Existence of Hierarchy Paths, [BHS15, Theorem 4.4]). Let pX ,Sq be hierar-chically hyperbolic. Then there exists D0 so that any x, y P X are joined by a D0-hierarchypath.

Whenever possible, one should work with hierarchy paths rather than other quasigeodesics,even actual geodesics. Unfortunately, not all quasigeodesics are hierarchy paths (meaningthat one cannot control how close the projection to some CW of a pD,Dq–quasigeodesic isto being a geodesic as a function of D only). In fact, there are spiraling quasigeodesics inR2, and, even worse than that, it is a folklore result that in mapping class groups there arequasigodesics that project to “arbitrarily bad” paths even in the curve graph of the wholesurface.

Moreover, hierarchy paths with given endpoints are not coarsely unique: think of R2,where there are plenty of quasigeodesics monotone in each factor that connect points faraway along a diagonal. In fact, it is a very important problem to study to what extent onecan make hierarchy paths canonical by adding more restrictions.

4.3. Realisation. In this subsection we discuss the realisation theorem, which says thatthe consistency inequalities characterise the tuples pπW pxqqWPS for x P X . We think of theπW pxq as the coordinates of x.

Definition 4.4 (Consistent). Fix κ ě 0 and let ~b Pś

UPS 2CU be a tuple such that for each

U P S, the coordinate bU is a subset of CU with diamCU pbU q ď κ. The tuple ~b is κ–consistentif, whenever V&W ,

min

dW pbW , ρVW q, dV pbV , ρ

WV q

(

ď κ

and whenever V Ď W ,

min

dW pbW , ρVW q, diamCV pbV Y ρ

WV pbW qq

(

ď κ.

(Notice that in the definition of consistent tuple we need the map ρWV for V Ĺ W .)

Theorem 4.5 (Realisation of consistent tuples, [BHS15, Theorem 3.1]). For each κ ě 1

there exist θe, θu ě 0 such that the following holds. Let ~b Pś

WPS 2CW be κ–consistent; for

each W , let bW denote the CW–coordinate of ~b.Then there exists x P X so that dW pbW , πW pxqq ď θe for all CW P S. Moreover, x is

coarsely unique in the sense that the set of all x which satisfy dW pbW , πW pxqq ď θe in eachCW P S, has diameter at most θu.

As mentioned in Part 1, the realisation theorem is used to perform constructions inall the CW separately and then pull those back to X . One such construction is (at last!)that of standard product regions. Basically, we fix U P S, and consider partial systems ofcoordinates pbV q, where we only assign bV when either V Ď U or V KU . If this partial systemof coordinates satisfies the consistency inequalities, we can extend it and use realisation tofind a corresponding point in X . The standard product region associated to U is obtainedconsidering all such realisation points. A similar game can be played starting from pairwiseorthogonal Ui, but for simplicity we stick to the case of a single U . Let us make this moreprecise.

Definition 4.6 (Nested partial tuple). Recall SU “ tV P S : V Ď Uu. Fix κ ě κ0 and letFU be the set of κ–consistent tuples in

ś

V PSU2CV .

Definition 4.7 (Orthogonal partial tuple). Let SKU “ tV P S : V KUu Y tAu, where A isa Ď–minimal element W such that V Ď W for all V KU . Fix κ ě κ0, let EU be the set ofκ–consistent tuples in

ś

V PSKU´tAu

2CV .


Construction 4.8 (Product regions in X ). Given X and U P S, there are coarsely well-defined maps φĎ, φK : FU ,EU Ñ X which extend to a coarsely well-defined map φU : FU ˆ

EU Ñ X , whose image PU we call a standard product region. Indeed, for each p~a,~bq P

FU ˆ EU , and each V P S, define the co-ordinate pφU p~a,~bqqV as follows. If V Ď U , then

pφU p~a,~bqqV “ aV . If V KU , then pφU p~a,~bqqV “ bV . If V&U , then pφU p~a,~bqqV “ ρUV . Finally,

if U Ď V , and U ‰ V , let pφU p~a,~bqqV “ ρUV .By design of the axioms, it is straightforward (but a bit tedious) to check that we actually

defined a consistent tuple, see [BHS17a, Section 13.1].

We notice that by the very definition of PU , the following hold. First, πY pPU q is uniformlybounded if U Ĺ Y (making sure that it makes sense to think of U as being coned-off to getCY ), as well as if U&Y (so that we can actually think of PU and PY as “independent”).

Coarse retractions onto standard product regions have been mentioned above. It shouldnot be hard to guess how they are constructed at this point. One simply starts with x P X ,define coordinates by taking πY pxq whenever Y Ď U or Y KU and ρUY otherwise, and takes arealisation point. Basically, one defines the retraction of x P X by keeping the coordinatesinvolved in the standard product region only.

Coarse median. As mentioned in Part 1, the realisation theorem can be used to construct acoarse-median map in the sense of [Bow13] (also called centroid in [BM11]). This is the mapm : X 3 Ñ X defined as follows. Let x, y, z P X and, for each U P S, let bU be a coarse centrefor the triangle with vertices πU pxq, πU pyq, πU pzq. More precisely, bU is any point in CU withthe property that there exists a geodesic triangle in CU with vertices in πU pxq, πU pyq, πU pzqeach of whose sides contains a point within distance δ of bU , where δ is the hyperbolicityconstant of CU .

By [BHS15, Lemma 2.6] (which is easy, and a good exercise), the tuple~b Pś

UPS 2CU whoseU -coordinate is bU is κ–consistent for an appropriate choice of κ. Hence, by the realisationtheorem, there exists m “ mpx, y, zq P X such that dU pm, bU q is uniformly bounded for allU P S. Moreover, this is coarsely well-defined, by the uniqueness axiom. The fact that thiscoarse median map actually makes the HHS into a coarse median space, and that, moreover,the rank is the “expected” one, is [BHS17c, Corollary 2.15].

The existence of the coarse median has many useful consequences, for example regardingasymptotic cones [Bow13, Bow15] ([BHS17c] heavily relies on these). Also, the explicitconstruction itself is useful in various arguments, for example to construct the kind ofretractions mentioned in the subsection on hierarchical quasiconvexity below.

5. Additional tools

5.1. Hierarchical quasiconvexity. Quasiconvex subspaces are important in the study ofhyperbolic spaces. The corresponding notion in the HHS world is hierarchical quasiconvexity.Prominent examples of hierarchically quasiconvex subspaces are standard product regions.

The natural first guess for what a hierarchically quasiconvex subspace of an HHS X shouldbe is a subspace that projects to uniformly quasiconvex subspaces in all CU . This is partof the definition, but not quite enough to have a good notion. In fact, the aforementionedproperty is satisfied by subspaces that are not even coarsely connected. There are at leasttwo ideas to “complete” the definition.

The first idea, and the one leading to the definition given in [BHS15], is that not only theprojections to the CU should be quasiconvex, but they should also determine the subspace.One ensures that this is the case by requiring that all realisation points of coordinatespbU P πU pQqq lie close to Q, where Q is the hierarchically quasiconvex subspace. That alsoensures that Q is an HHS itself, see [BHS15, Proposition 5.6].


A picture that one should keep in mind regarding this second property is that it is notsatisfied by an ”L” in R2, even though its projections to the two factors are convex. Rather,what one wants is a ”full” square. This is related to the difference, in the cubical world,between `1–isometric embeddings (the ”L”) as opposed to convex embeddings (the square).

The second idea to complete the definition is probably more intuitive. Unfortunately, asof yet it has not been proven that this gives an equivalent notion. Recall that given an HHS,there is a preferred family of quasigeodesics connecting pairs of points, which are calledhierarchy paths, and that their defining property is that they project to unparametrizedquasigeodesics in all CU . It is then natural to just replace geodesics in the definition of theusual quasiconvexity in hyperbolic spaces with hierarchy paths. Namely we want to say thatQ is hierarchically quasiconvex if all hierarchy paths joining points of Q stay close to Q. It iseasy to see that this implies that the projections of Q to the CU are quasiconvex, but theadditional property is not clear, and it would be interesting to known whether it is satisfied.It is however true (and easy to see) that hierarchy paths joining points on a hierarchicallyquasiconvex set stay close to it. Moreover, the hull of two points, as defined in the nextsubsection, coarsely coincides with the union of all hierarchy paths (with fixed, large enoughconstant) connecting them.

Definition 5.1 (Hierarchical quasiconvexity). [BHS15, Definition 5.1] Let pX ,Sq be ahierarchically hyperbolic space. Then Y Ď X is k–hierarchically quasiconvex, for somek : r0,8q Ñ r0,8q, if the following hold:

(1) For all U P S, the projection πU pYq is a kp0q–quasiconvex subspace of the δ–hyperbolicspace CU .

(2) For all κ ě 0 and κ-consistent tuples ~b Pś

UPS 2CU with bU Ď πU pYq for all U P S,each point x P X for which dU pπU pxq, bU q ď θepκq (where θepκq is as in Theorem 4.5)satisfies dpx,Yq ď kpκq.

Remark 5.2. Note that condition (2) in the above definition is equivalent to: For everyκ ą 0 and every point x P X satisfying dU pπU pxq, πU pYqq ď κ for all U P S, has the propertythat dpx,Yq ď kpκq.

A very important property of hierarchically quasiconvex subspaces is that they admit anatural coarse retraction, which generalises the retraction onto standard product regions.The definition (see [BHS15, Definition 5.4]) is that a coarsely Lipschitz map gY : X Ñ Yis called a gate map if for each x P X it satisfies: gYpxq is a point y P Y such that for allV P S, the set πV pyq (uniformly) coarsely coincides with the projection of πV pxq to thekp0q–quasiconvex set πV pYq.

The uniqueness axiom implies that when such a map exists it is coarsely well-defined, andthe existence is proven in [BHS15, Lemma 5.5].

5.2. Hulls and their cubulation. Other important examples of hierarchically quasiconvexsubspaces are hulls. In a δ-hyperbolic space X, if one considers any subset A Ď X, then onecan construct a “quasiconvex hull” simply by taking the union of all geodesics connectingpairs of points in the space. It is easy to show that this is a 2δ-quasiconvex subspace, whateverA is. In an HHS, we can proceed similarly.

Definition 5.3 (Hull of a set; [BHS15]). For each A Ă X and θ ě 0, let the hull, HθpAq, bethe set of all p P X so that, for each W P S, the set πW ppq lies at distance at most θ fromhullCW pAq, the convex hull of A in the hyperbolic space CW (that is to say, the union of allgeodesics in CW joining points of A). Note that A Ă HθpAq.

It is proven in [BHS15, Lemma 6.2] that for each sufficiently large θ there exists κ : R` Ñ R`so that for each A Ă X the set HθpAq is κ–hierarchically quasiconvex.


To connect a few things we have seen so far, the gate map onto the hull of two points x, ycoincides with taking the median mpx, y, ¨q.

Hulls of finite sets carry more structure than just being hierarchically quasiconvex. In ahyperbolic space, hulls of finitely many points are quasi-isometric to trees, with constantsdepending only on how many points one is considering. Trees are 1-dimensional CAT(0) cubecomplexes, and what happens in more general HHSs is that hulls of finitely many points arequasi-isometric to possibly higher dimensional CAT(0) cube complexes:

Theorem 5.4. [BHS17c, Theorem C] Let pX ,Sq be a hierarchically hyperbolic space andlet k P N. Then there exists M0 so that for all M ě M0 there is a constant C1 so thatfor any A Ă X of cardinality ď k, there is a C1–quasimedian2 pC1, C1q–quasi-isometrypA : Y Ñ HθpAq.

Moreover, let U be the set of U P S so that dU px, yq ěM for some x, y P A. Then dimYis equal to the maximum cardinality of a set of pairwise-orthogonal elements of U .

Finally, there exist 0–cubes y1, . . . , yk1 P Y so that k1 ď k and Y is equal to the convex hullin Y of ty1, . . . , yk1u.

The theorem above is crucial for the proof of the quasiflats theorem in [BHS17c], and itdefinitely feels like it should have many more applications.

5.3. Factored spaces. In this section we discuss a construction where one cones-off subspacesof an HHS and obtains a new HHS. Examples of HHSs that are obtained using this constructionfrom some other HHS are the “main” hyperbolic space CS, which is obtained coning-off standard product regions in the corresponding X , and the Weil-Petersson metric onTeichmuller space, which is (coarsely) obtained coning-off Dehn twist flats in the correspondingmapping class group. The construction was devised in [BHS17b], with the purpose of havingspaces that “interpolate” between a given HHS X and the hyperbolic space CS. In particular,this gives one way of doing induction arguments.

Given a hierarchical space pX ,Sq, we say U Ď S is closed under nesting if for all U P U, ifV P S´ U, then V Ď U .

Definition 5.5 (Factored space). [BHS17b, Definition 2.1] Let pX ,Sq be a hierarchically

hyperbolic space. A factored space pXU is constructed by defining a new metric d on Xdepending on a given subset U Ă S which is closed under nesting. First, for each U P U,for each pair x, y P X for which there exists e P EU such that x, y P FU ˆ teu, we setd1px, yq “ mint1, dpx, yqu. For any pair x, y P X for which there does not exists such

an e we set d1px, yq “ dpx, yq. We now define the distance d on pXU. Given a sequence

x0, x1, . . . , xk P pXU, define its length to beřk´1i“1 d1pxi, xi`1q. Given x, x1 P pXU, let dpx, x1q be

the infimum of the lengths of such sequences x “ x0, . . . , xk “ x1.

It is proven in [BHS17b, Proposition 2.4] that factored spaces are HHSs themselves, withthe “obvious” substructure of the original HHS structure.

I believe that factored spaces will play an important role in studying quasi-isometriesbetween hierarchically hyperbolic spaces in view of [BHS17c, Corollary 6.3], which says thata quasi-isometry between HHSs induces a quasi-isometry between certain factored spaces.One can then take full advantage of the HHS machinery to iterate this procedure and getmore information about the quasi-isometry under examination. This strategy should workin various sub-classes of right-angled Artin and Coxeter groups (one example is given in[BHS17c]).

2CAT(0) cube complexes are endowed have a natural median map. Here by C1–quasimedian we mean thatthe coarse median of 3 points is mapped within distance C1 of the median of their images.


5.4. Boundary. HHSs admit a boundary, defined in [DHS17]. Here is the idea behind thedefinition. We have to understand how one goes to infinity in an HHS, and more precisely wewant to understand how one goes to infinity along a hierarchy ray, because we like hierarchypaths. The hierarchy ray will project to any given CU close to either a geodesic segment or ageodesic ray. In fact, a consequence of large links is (the hopefully intuitive fact) that theremust be some CU so that the hierarchy ray has unbounded projection there (see the useful“passing up” lemma [BHS15, Lemma 2.5]). Now, one can consider the non-empty set of all CUwhere the hierarchy ray has unbounded projection, and it should come to no surprise thatthey must be pairwise orthogonal, because in all other cases there are constraints comingfrom the consistency inequalities on the projections of points of X . As a last thing, youmight want to measure how fast you are moving asymptotically in each of the CU and recordthe ratios of the various speeds. These considerations lead to the following definition.

Definition 5.6. [DHS17, Definitions 2.2-2.3] A support set S Ă S is a set with SiKSj for

all Si, Sj P S. Given a support set S, a boundary point with support S is a formal sump “

ř

SPS apSpS , where each pS P BCS, and apS ą 0, and

ř

SPS apS “ 1. (Such sums are

necessarily finite.)The boundary BpX ,Sq of pX ,Sq is the set of boundary points.

The boundary also comes with a topology, which is unfortunately very complicated todefine. But fortunately, for at least some applications one can just use a few of its propertiesand never work with the actual definition (this is the case for the rank rigidity theorems[DHS17, Theorems 9.13,914]). The main property is that if an HHS is proper as a metricspace then its boundary is compact by [DHS17, Theorem 3.4]. Another good property isthat hierarchically quasiconvex subspaces (and not only those) have well-defined boundaryextensions.

5.5. Modifying the HHS structure. There are various way to modify an HHS structure,and this is useful to perform certain constructions. For example, there is a combinationtheorem for trees of HHSs which, unsurprisingly, requires the HHS structures of the variousedge and vertex groups to be “compatible”. In concrete cases, one might be starting with atree of HHSs that does not satisfy the compatibility conditions on the nose, but does aftersuitably adjusting the various HHS structures. This is the case even for very simple exampleslike the amalgamated product of two mapping class groups over maximal virtually cyclicsubgroups containing a pseudo-Anosov.

A typical way of modifying the HHS structure is to cone-off a collection of quasi-convexsubspaces of one of the CU , and then add those quasi-convex subspaces as new hyperbolicspaces. This usually amounts to considering certain (hierarchically quasiconvex and hyper-bolic) subspaces of X as product regions with a trivial factor. This construction is usedto set things up for the small cancellation constructions in [BHS17b], and is explored inmuch more depth in [Spr]. In the latter paper the author studies very general families ofquasiconvex subspaces of CS (called factor systems) so that one can perform a version of thecone-off-and-add-separately construction mentioned above. This is needed to set things upbefore using the combination theorem in natural examples, like the one mentioned above.

In another direction, one may wonder whether there exists a minimal HHS structure, andin particular one might want to reverse the procedure above. This is explored in [ABD17].

References

[ABD17] Carolyn Abbott, Jason Behrstock, and Matthew Gentry Durham. Largest acylindrical actions andstability in hierarchically hyperbolic groups. arXiv:1705.06219, 2017.

[BBF15] Mladen Bestvina, Ken Bromberg, and Koji Fujiwara. Constructing group actions on quasi-trees

and applications to mapping class groups. Publ. Math. Inst. Hautes Etudes Sci., 122:1–64, 2015.


[Beh06] Jason A. Behrstock. Asymptotic geometry of the mapping class group and Teichmuller space. Geom.Topol., 10:1523–1578, 2006.

[BHS15] Jason Behrstock, Mark F Hagen, and Alessandro Sisto. Hierarchically hyperbolic spaces II: combi-nation theorems and the distance formula. arXiv:1509.00632, 2015.

[BHS17a] Jason Behrstock, Mark Hagen, and Alessandro Sisto. Hierarchically hyperbolic spaces, I: Curvecomplexes for cubical groups. Geom. Topol., 21(3):1731–1804, 2017.

[BHS17b] Jason Behrstock, Mark F Hagen, and Alessandro Sisto. Asymptotic dimension and small-cancellationfor hierarchically hyperbolic spaces and groups. Proc. Lond. Math. Soc., 114(5):890–926, 2017.

[BHS17c] Jason Behrstock, Mark F Hagen, and Alessandro Sisto. Quasiflats in hierarchically hyperbolicspaces. arXiv:1704.04271, 2017.

[BM11] Jason A. Behrstock and Yair N. Minsky. Centroids and the rapid decay property in mapping classgroups. J. Lond. Math. Soc. (2), 84(3):765–784, 2011.

[Bow13] Brian H. Bowditch. Coarse median spaces and groups. Pacific J. Math., 261(1):53–93, 2013.[Bow15] Brian H Bowditch. Large-scale rigidity properties of the mapping class groups. preprint, 2015.[DHS17] Matthew G Durham, Mark F Hagen, and Alessandro Sisto. Boundaries and automorphisms of

hierarchically hyperbolic spaces. To appear in Geometry & Topology, arXiv:1604.01061, 2017.[Hae16] Thomas Haettel. Hyperbolic rigidity of higher rank lattices. arXiv:1607.02004, 2016.[HS16] Mark F Hagen and Tim Susse. Hierarchical hyperbolicity of all cubical groups. arXiv:1609.01313,

2016.[Hua17] Jingyin Huang. Top-dimensional quasiflats in CAT(0) cube complexes. Geom. Topol., 21(4):2281–

2352, 2017.[MM99] Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. I. Hyperbolicity. Invent.

Math., 138(1):103–149, 1999.[MM00] H. A. Masur and Y. N. Minsky. Geometry of the complex of curves. II. Hierarchical structure.

Geom. Funct. Anal., 10(4):902–974, 2000.[Mou16] Sarah C Mousley. Non-existence of boundary maps for some hierarchically hyperbolic spaces.

arXiv:1610.07691, 2016.[Mou17] Sarah C Mousley. Exotic limit sets of Teichmuller geodesics in the HHS boundary. arXiv:1704.08645,

2017.[Spr] Davide Spriano. Non-trivial hierarchically hyperbolic structures on hyperbolic spaces and applica-

tions. In preparation.[ST16] Alessandro Sisto and Samuel J. Taylor. Largest projections for random walks. To appear in Math.

Res. Lett., arXiv:1611.07545, 2016.

Department of Mathematics, ETH Zurich, 8092 Zurich, SwitzerlandE-mail address: [email protected]

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The rst part of this survey is a heuristic, non-technical ... · The rst part of this survey is a heuristic, non-technical discussion of what an HHS is, and the aim is to provide